Maybe you are trying too hard, have you tried the flavored milk, choc/ strawberry /banana and it wont hurt him to leave milk alone for a while, at 3 year old I am assuming he eats fruits
and veg
and protein , children like pasta , rice, beans,give him plenty of fresh water and some juice.Sometimes we worry more than we have to about children eating, if we make more of it more it becomes . So back off let him eat his food and in a very short time he will drink milk again .